and Applied Analysis 3 Definition 1.4 see 18, 20–23 . 1 The intersection of all fuzzy α-closed resp., semiclosed, preclosed, γ-closed, semi-preclosed, β-closed sets containing a fuzzy set A is called a fuzzy α-closure resp., semiclosure, preclosure, γ-closure, semi-preclosure, β-closure of A. 2 The union of all fuzzy α-open resp., semiopen, preopen, γ-open, semi-preopen, β-open sets contained i...

In this paper, a collection of sets are investigated in such way that the splits preopen and b-open also semi-open sets. Functions terms is part paper it will be considered these remains invariant under homeomorphic image.

The purpose of this paper is to define and study some new types hesitant fuzzy open sets namely, α -open, preopen, semiopen, b-open β -open in topological space. Some properties the relationships between these are investigated. Furthermore, them subspace introduced.

The concepts of P-compactness, countable P-compactness, the P-Lindelöf property are introduced in L-topological spaces by means of preopen L-sets and their inequalities when L is a complete DeMorgan algebra. These definitions do not rely on the structure of the basis lattice L and no distributivity in L is required. They can also be characterized by means of preclosed L-sets and their inequalit...

-open sets were introduced and studied by Janković and Hamlett (1990) to generalize the well-known Banach category theorem. Quasi-openness was introduced and studied by Abd El-Monsef et al. (2000). These are∗-dense-in-itself sets of the ideal spaces. In this note, properties of these sets are further investigated and characterizations of these sets are given. Also, their relation with -dense se...